A single error correcting code, Long(6, 3), is designed so that 3 parity check digits are...

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A single error correcting code, Long(6, 3), is designed so that 3 parity check digits are appended to the end of a 3-digit input, using GF(7). The parity check matrix H is given by 1 1 1 H = 1 2 3 1 1 1 4 5 6 1 2 3 4 5 6 1 1 1 1 1 1 1 2 3 4 5 6 1 4 2 2 4 1 (i) Derive the equations to generate the 3 parity check digits from the parity check matrix H, briefly explaining the working steps. 10 marks When the input is 123, what are the check digits? 3 marks (iii) Explain how you would use the coefficients in the parity check matrix to detect and correct a single error. 4 marks For the following received codewords, determine whether they have any errors, and if so correct them if possible. 655233, 246135, 205123 10 marks (iv) Describe the similarities and differences between this code and the Ham (7,4) code in terms of their implementation and application. 5 marks A single error correcting code, Long(6, 3), is designed so that 3 parity check digits are appended to the end of a 3-digit input, using GF(7). The parity check matrix H is given by 1 1 1 H = 1 2 3 1 1 1 4 5 6 1 2 3 4 5 6 1 1 1 1 1 1 1 2 3 4 5 6 1 4 2 2 4 1 (i) Derive the equations to generate the 3 parity check digits from the parity check matrix H, briefly explaining the working steps. 10 marks When the input is 123, what are the check digits? 3 marks (iii) Explain how you would use the coefficients in the parity check matrix to detect and correct a single error. 4 marks For the following received codewords, determine whether they have any errors, and if so correct them if possible. 655233, 246135, 205123 10 marks (iv) Describe the similarities and differences between this code and the Ham (7,4) code in terms of their implementation and application. 5 marks